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Ok, I have warned you. Brace yourself. I am going to argue against what many believe to be the most fundamental law of physics. A sacrosanct law. A law that you experience everyday and that is so obviously true that no one should meddle with it. You all know this law. It's the law referred to as 'the second'. The second law of thermodynamics.

Ok, don't blow it hammock guy. You can argue against every concept in physics, but don't touch the second law of thermodynamics! You don't want to place yourself in this group, do you?

Has anyone ever dared to attack the second law of thermodynamics? I am not aware of even a single attempt. Eddington's famous remark on the second law of thermodynamics says it all. This law occupies a special position in physics and rules supreme. Beware the person who dares to meddle with it:

It was Rudolf Clausius, the inventor of the term 'entropy', who already told us that the entropy of the universe* can go only one direction: it can only increase. And once Ludwig Boltzmann and Claude Shannon were done, we became familiar with the true meaning of the term entropy, and it all started to make sense. Entropy simply is the information content of a physical system, the number of digits needed to fully describe the system down to its smallest details. So the second law tells us that the number of digits needed to describe the universe keeps increasing. Seems fair enough, right? The whole universe expands, so probably its information content increases as well.

One-way Universe

Wait a second. Take Newton's laws, or any other fundamental law of physics. All these laws are reversible. They describe a two-way universe. Look at two elastically colliding billiard balls. Now consider the same collision reversed in time. This reverse collision is as valid as the collision going forward in time. Our laws of physics describe a two-way universe. A universe without an 'arrow of time'. This holds for all fundamental dynamical laws, including the las that describe the dynamics of atomic and subatomic particles.

Yet, our human experience that focuses on scales much larger than microscopic scales is vastly different. We seem to live in a one-way universe. Glass can brake, but it can't unbreak. You can turn a piece of wood into ashes, but no one has managed to reverse that process. We grow older, never younger. We all can remember the past but not the future.

What is the origin of this arrow of time? Why does time present itself as a one-way road?

Feynman contemplated these issues, and considered them of enough relevance to bring to the attention of undergraduate students at Caltech. In his 'Feynman Lectures on Physics' he points towards the big bang as the cause of the one-way-ness:

Many years later another Caltech physicist and science communicator, Sean Carroll, continues to study the puzzle of entropy increase breaking the two-way nature of our universe:

So the current consensus is that entropy increases simply because there are many more configurations that require a large number of digits to describe them, then that there are configurations described by fewer digits. We all now that with four digits you can describe a total of 10,000 configurations, with two digits only 100. So statistically, a physical system will be likely to end up in configurations described by more digits. But as Sean Carroll stresses, that still leaves the question why the universe started in such a particular low entropy configuration described by very few digits.

Since his 1989 book 'The Emperor's New Mind', Roger Penrose has been drawing the attention of the wider public to the remarkable low entropy of the big bang. In his magnum opus 'The Road to Reality' he revisits the issue, and finally in his latest book 'Cycles of Time' he presents a whole new cosmology aimed at explaining the low entropy beginning of our universe.

In my last blog post I discussed 'Cycles of Time' and made the statement that the big bang must have had a low entropy, but that very fact should not surprise us. The best way to explain this is by studying a simple toy model.

Fibonacci Universe

Imagine a universe much simpler than ours. A universe in which time progresses in discrete steps. At each tick this universe jumps into a new spatial configuration. The physicists living in this universe have managed to describe these configurations, down to their most minute details, in terms of simple numbers. They discover that at a certain step their universe is in a configuration that can be described by the number 4,298,034,510. One tick later the universe is in configuration 6,954,365,922. Then in 11,252,400,432, and next in 18,206,766,354. Scientists study these numbers and try to find a pattern. "If we find the mathematical law that generates these numbers, we will have a theory of everything" they exclaim. "We will know the mind of God!"

They readily observe that all numbers are even, and coin this 'the first law'. They also recon that the numbers keep growing, this provides them with their 'second law'. Next they discover how to quantify this growth: each next number appears to be about 1.618034 times the previous number.

Based on experiments in their particle accelerators the scientists start contemplating how the Second Law of configuration number growth would apply to an anti-universe obtained by replacing all particles by anti-particles. It is not before long that theorists come up with a growth rule for anti-universes: each next configuration number in an anti-universe will be -1.618034 times the previous number. The negative sign causes negative configuration numbers to appear, but the second law will still hold as the configuration numbers would not just oscillate but also grow in size. There is a lot of discussion about the meaning of the minus sign in the growth number. However, this discussion subsides when the scientists start realizing that growth laws can only be an approximation to a deeper truth. Simple reason being that a repetitive multiplication by a non-integer number can't produce pure integers. Many scientists frantically try to find out how exactly each integer determines the next. No one succeeds.

Then a young scientist starts toying with the idea that a law that determines the next configuration based solely on the previous one might not exist. Could it be that two or more subsequent configurations are needed to determine the next? He immediately stumbles upon an amazingly simple pattern in which each number is nothing more the sum of the previous two numbers.

This hits the scientific community like a bomb shell.The scientist now realize they were pursuing the wrong approach by assuming the state of the universe requires only one configuration number. So they start considering pairs of subsequent configuration numbers as describing the universe:

It is as if the number of dimensions of the universe they live in have doubled overnight! For reasons no one remembers, the scientist start referring to the space of configurations described by pairs of numbers as the phase space. This to distinguish it from the configuration space that carries only a single number.They next discover that the anti-universe to their own universe is obtained simply by changing the sign of one of the numbers in the phase-space pair. So

would describe the anti-universe of

and vice versa. By applying the 'add-the-last-two' rule to the numbers appearing in subsequent pairs, they can reproduce the observed evolution in terms of the configuration pairs:

and they can do the same for the anti-universe:

A new riddle has appeared. Why do a universe and its anti-partner evolve differently? This can't be right as universes and their anti-universes are related by a simple sign change in the configuration pairs.

It doesn't take long before a bright mind realizes a universe and its anti universe both evolve according to the 'add-the-last-two' rule, but they do so in opposite directions in time.

This means that by knowing a configuration pair, it is not only possible to compute forward in time, but also to trace back the history of their universe. And the amazing thing is: this can be done by using the very same 'add-the-last-two' rule, provided one changes the configurational pair into the corresponding anti-pair and reverses the order of the two numbers in the pair (an operation referred to as time reversal):

Now all pieces of the puzzle have fallen in place. By transforming universes in their anti universes and back again, the scientists can compute the phase-space state of the universe forward and backward in time. They can now answer the mother of all questions "how did it all start?". So they begin computing back in time. Starting from the pair

they reach lower and lower numbers:

No small task given the rudimentary computational resources in their rather limited universe. But they keep calculating and reach still lower numbers:

The computations continue further. And further. The numbers reduce in size to 3-digit and then 2-digit numbers. Then something remarkable happens, the numbers start growing in size:

Did they make an error? They check their calculations, and check them again. Everything seems right. What is happening here? Have they stumbled upon an exception to the second law? If the numbers grow in size when going backwards in time, they shrink going forward in time. But the second law forbids this. And this law rules supreme. Or does it?

The conclusion seems inevitable: if the second law of thermodynamics doesn't hold true for any of these simple model systems, it is probably neither true for our own universe. One could argue that all of this just means the second law of thermodynamics holds true only at one side of the bounce. That would be compatible with the observed entropy increase following the big bounce that we refer to as the big bang.

This is a too restrictive view. It is much more insightful to start from a timeless description and to view an unbound reversible dynamics as a chain of states that never retraces its steps and that therefore has no beginning and no end. In terms of the length of the description of the states (the entropies of the states), such a chain can not be monotonic. The entropy of the states has to feature a bounce. This is inevitable, simply because the length of the descriptions can't drop below zero. With a bounce, the direction of entropy increase at either side of the bounce is pointed away from the bounce. And as low entropy states can not carry the memory of high entropy states, a temporal direction with an unknown future results.**

Where does this leave us with the Second Law? Clearly when expressed a-la Clausius "The entropy of the universe can only increase" does not hold true. However, we can redefine the Second Law such that it does cover the observed bounce behavior:

"The entropy of the universe is a convex function of time."

With S(X) denoting the entropy of the universe in state X, Clausius formulaition of the second law reads:

if B comes after A

The above findings suggest a replacement of the form:

for any states A and B

Is this the correct formulation of the second law? I don't know. But I do know that when considering model systems, this convexity formulation makes more sense than the standard formulation. Also this new formulation, in contrast to Clausius' formulation, makes no use of an ordering of events, and fits naturally with reversible laws of physics.

An important corollary of a convex function or bounce-description of the second law is that one should not wonder about the smallness of the entropy of our universe about 13.7 billion years ago. If the universe does not retrace its steps, it has to have a minimum entropy at some point. That point we refer to as the big bang, but that should be more appropriately be referred to as the big bounce.

Assuming that the big bang represents a temporal singularity that marks the start of time is an unnecessary assumption. An assumption that gets you in all kind of trouble and forces the question why the starting point was so special and had such a low entropy.

A big bounce seems to me a cleaner description that more easily survives Occam's razor than a big bang singularity. Let's remind ourselves of the fact that there is no single piece of evidence that would make us believe that our universe is in any way bounded, either in spatial extend or in time. For all we know, the universe will keep expanding indefinitely in all directions. Similarly, the universe can extend in both time directions without limit. There is no reason to believe that our universe won't reach everywhere and everywhen.

Only when we know the ultimate theory, will we know for sure the character of the state referred to as the big bang. In the meantime, you have to make a choice in what answer you are going to give to the question "what was there before the big bang?". Will your answer be the traditional

A) "This is a nonsense question, you might as well have asked what is south of the South Pole!".

Or will it rather be a bounce-inspired

B) "An universe that is a mirror universe to our own, consisting of anti-particles moving back in time and mirrored in space."

Faites vos jeux.

Notes

* Throughout this blog post, wherever the term 'our universe' appears, this refers to our observable universe.

** I am making some big jumps here that deserve a separate blog post. Note that I am not making any statement on how the laws of physics can lead to memory in conscious beings. No one knows how to make that link.

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Comments

Strictly speaking the second law only applies to parts of processes that have the energy randomly distributed. For biased processes you need the third law as presented by Schrödinger in his book "WHAT IS LIFE."

Johannes opened this topic before with a challenge for perpetual motion machines. Jerry responded with two examples that are easily provable and are well documented. One was dark current in a diode. The other was radiant focusing in a parabolic reflector.

In this second example, the target becomes hotter than the source, with no work done and no heat given off to a lower temperature. It is a proven violation of the second law. The third law governs it.

The first example is also a violation of the second law. Energy flows without a potential difference, governed by diffusion acrost the diode junction.

In Jerry's book he developes the radiant focusing as a major topic to answer questions about how order originated in the universe.

In the meantime, you have to make a choice in what answer you are going to give to the question "what was there before the big bang?". Will your answer be the traditional A) ... Or will it rather be a bounce-inspired B) ...

Inflation addresses the very early universe, it has little (if anything) to say about the character of the singularity in the classical picture. The question: "is this singularity spacelike and can it therefore be smoothly extended into pre big bang stages", is a yes/no question.

The second law(Entropy flat or rising) only applies to closed systems
1) One could debate whether the expanding Universe is closed or open
2) In the model of 2 mirrored universes, our universe is open(it opens into the mirror), so the second law does not apply to our universe in this model
Only in a closed case(either 1 closed universe or 1 closed set of 2 mirrored universes) the sesond law does apply to my conviction

These two points ndeed deserve some further explanation.
1) To decide if a system is open or closed, you just have to ask yourself the question whether the system contains all the information to evolve its configurations forward and backward in time.
2) The trick here is to take a step back and study the totality (universe + it's pre-big-bang anti mirror image). That whole system is closed. (The same applies mutatis mutandis to the expnding universe.)

You can turn ashes back to wood. Forests burn down, they grow back. It is a cycle that has been taking place for millions of years. Order is subjective. Maybe you consider order to be the deck of cards to be in a nice, neat stack, and I consider order to be the cards flung a cross the room. Both are states from which any changes could be claimed to be disorder. We could decide that some state of the universe far in the future is the definition of order, and as we race towards that point in time, the universe is spontaneously organizing itself into that state.

"You can turn ashes back to wood. Forests burn down, they grow back."
Sure, but that is cheating... In your scenario you are not reversing the process of burning. Instead you (or rather nature) starts from tree seeds, sunlight, etc. to regrow a forest. These low entropy entities that you start off with are not the product of the proces of burning.
If I burn down a tree and give you all the products that result (ashes, CO2, heat, etc.) can you turn these back into a tree?

If the minimal entropy position is maximally minimal then aren't the phase states heading in and out the same, thus they could be viewed as the same states.
So you bounce off and head back the way you came.

That's a good question that I should dicuss in another blog post.
The short answer is: the anti-universe evolving at 'the other side of the big bang' could be the exact PCT (Parity Charge and Time) reversed image of our universe, but it need not be. Personally I think it would be most elegant if it was, but nature is probable not bothered about the wishes of some guy in a hammock. If it is the exact mirro image, than indeed this would be a nice way to satisfy strictly reversible (unitary) dynamics without introducing any 'additional universes.

so you says entropy is the measure of everything. So that would included all the things in my mind. So when I die there must be a loss of some stuff. Unless I am say reborn in some other part of the universe. Sorry hope you didn't fall out of your hammock.

"when I die there must be a loss of some stuff. Unless I am say reborn in some other part of the universe."
When you die, there will no doubt be people who feel a loss. From a physics point of view, however, there is no loss. All information is retained. The information that makes 'you' lives on albeit in diluted form and no longer accessible to us.
Perhaps you need a hammock now? ;)

"The information that makes 'you' lives on albeit in diluted form and no longer accessible to us" if so how can you measure it ? and if you cannot measure it how can it be accounted for in the equation?

Exactly. It's an information loss (but, then, so is the act of forgetting where you put your car keys) and I think it's not quite on the scale of "burning a forest down." Nor is there a rebound effect.

You cite Claude Shannon, but his definition of entropy doesn't fit with yours -- it's a measure for the uncertainty in a message. The equation systems for both concepts are fairly different, so to my eye it's sort of like comparing apples and orangutangs.

Is heat analogous to information? Is irreversible heat (friction) similar to disordered information? If you add energy to a system, you can't reclaim lost heat, just add new heat. If you add energy to a system of disordered information, you can reclaim lost information. It seems to me that heat can be lost, but information can be reclaimed.

In the field of Information Science, you can add filters to reduce noise... but (from the standpoint of the papers I read and from the standpoint of classic information science.) You can also add formulas to extrapolate information (you see a black pixel here and one space away another black pixel and if you're dealing with a stream of information that represents (say) a picture you can assume that most of the time the information between the two will represent a black pixel.)

But you're adding a lot of energy to re-order the decrease of entropy.

I myself truly and instinctively believe that everything I have read here, including the comments are treating this system (our universe) like it is a philosophical debate or some kind of metaphysical argument! I strongly believe that a comparing the supposed entropy of the universe to simple fibonacci numbers or any numerical set is completely wrong, un-insightful and doomed to failure-we all know that quantumn mechanics and relativity dont like each other-we are still missing something HUGE, or simple, i dont know, but I do know we arent going to find the answer fiddling with silly numbers on a piece of paper with a human mind at the helm.....best bet...wait for some real data from LHC...and I mean wait...a few years at least!...its like a bunch of ants debating the size of the earth for gods sakes....

Absolutely, from Fibonacci numbers to the real universe is a loooong shot.

Yet, if all unbounded reversible systems generate a bounce in the entropy then the real thing (our universe) might very well be doing the same. All fundamental laws of physics describe a reversible universe, and it would be a huge surprise if the ultimate laws would not be reversible.

Big Bounce cosmologies are not new. Martin Boyowald (referred to in the blog) obtains such bounces from a discrete Loop Quantum Gravity description of the universe. That is still only a model system, but certainly closer to the truth than Fibonacci numbers. See this toying with Fibonacci numbers as a didactical exercise. I could have shown more complicated models all following the same bounce behavior.

We probably are like ants thinking about the size of the earth. But it's the only thing we have. Waiting for LHC results is not an option. In this ant-analogy the LHC is a standard yardstick: an enormous instrument from an ant's perspective, yet of little help in determining the size of the earth.

B) "An universe that is a mirror universe to our own, consisting of anti-particles moving back in time and mirrored in space."
Universe pairs of this nature have been intuitively obvious for several decades but nobody seems to have thought further - each would exist in each others' past and leave an effect - or the timelines isolated by more than the arrow direction - or the big bang/bounce wipes out history and entanglement of the first chronons becomes redundant.
Universe pairs of great number might occur simultaneously by a single brane banging event producing a salt crystal lattice of adjacent universes wherein the time orientation of adjacent pairs may or may not be in phase - re dark energy replaced by external gravitation.
History has consistently replaced singularities with multiplicities
Well done

I read in Wikipedia that a "system" is a matter of choice. So the universe being designated as an isolated system seems to be a matter of choice. The designation of boundaries also seems to be a matter of choice. So the applicability of the Second Law also seems to be a matter of choice.

I do not understand the reversibility of the collision of two billiard balls. If in the original collision one ball is accelerated from a cue and strikes the second ball, etc., the two balls will end up stationary. If you reverse the process the balls will accelerate toward each other by themselves, one will end up stationary and the other will end up pushing the cue backwards, etc. This does not seem to illustrate "reversibility" any more than does a piece of glass that shatters into different pieces after being struck by a hammer. What part of the collision is supposed to be reversible, and what kind of reversibility is it? (Say the cue ball was hit so hard it smashed the second ball into pieces.)

Is the "arrow of time" supposed to be a physically real "objective" phenomenon, or is it a non-physical subjective concept? It seems to me that a ray of light entering our eye can be said to arrive from the future; then the "arrow of time" would be from future to past. On the other hand, the ray of light being emitted at some point in some past of some source could be said arrive from the past. This seems to be a kind of relativity. So it seems to be a subjective judgement whether the arrow of time moves from past to future or future to past. I also doubt that "time" ticks discretely like a clock.

It seems that the "Big Bang" idea rests on the assumption that there is a constant amount of mass-energy in the universe, and running the film backward produces a hot dense state. But I don't believe this is necessarily so. The universe may be a system for making mass, in which case there would be less remembered mass and more mass to come (from the future). The CMB would then be emitted by newly created hydrogen plasma way out there in the future.

"I do not understand the reversibility of the collision of two billiard balls. [..] What part of the collision is supposed to be reversible, and what kind of reversibility is it?"

I wrote "perfectly elastic collison of billiard balls". This is an idealization that is actually approached by real billiard balls during the time interval just before till just after collision. So it is the collision itself as analyed by Newton's laws. Think about two billiard ball approaching each other at constant speed, bouncing off, and continuing at constant speed. The reverse process is as feasible as the forward process.

The reversibility is then a very special case. Even in an irreversible thermodynamic process the collisions are reversible. So it seems to me that a special or micro-scale (space or time) case (universally applicable, however) is being applied to the universe as an arbitrarily chosen system, which is not the same as being universally applicable.

Ultimately the whole topical thrust seems to come from the assumption that our constant-energy universe (but a non-understood expanding one) is cooling down from a "big bang" (or bounce). In the irreversible thermodynamic "hot-to-cold" process, the micro-scale conservation laws are upheld in collisions between atoms and molecules of the gas, and the arbitrarily chosen system follows the second law. But who can point to a law that states the universe is cooling down from a "big bang" (or "bounce")?

Say the anti-universe (with one negative number in the "add-the-last-two" scenario) bounces to become a pro-universe (with no negative numbers). Does that anti-universe exist as an anti-twin of ours at this very moment? Then there are four universes, if our " two pro number" universe is half-mirrored by a universe with one pro- and one anti-number! If not, where does the single anti-universe come from? It cannot come from this universe which has no negative numbers. So if the anti-universe is the answer to the "mystery" of our existence, it doesn't seem any more satisfactory than "God", or "the South Pole".

"In the meantime, you have to make a choice in what answer you are going to give to the question "what was there before the big bang?". . . .

B) "An universe that is a mirror universe to our own, consisting of anti-particles moving back in time and mirrored in space."

Would that universe be getting smaller or larger, and in whose reference frame?

"Even in an irreversible thermodynamic process the collisions are reversible."

That is the whole point. At microscale there is no irreversibility. Yet at large scales we do observe irreversible effects. Since Boltzmann we know there is no contradiction.

"So it seems to me that a special or micro-scale (space or time) case (universally applicable, however) is being applied to the universe as an arbitrarily chosen system, which is not the same as being universally applicable. Ultimately ..."

Not sure what you want to say here.

"In the meantime, you have to make a choice in what answer you are going to give to the question "what was there before the big bang?". . . .B) "An universe that is a mirror universe to our own, consisting of anti-particles moving back in time and mirrored in space." Would that universe be getting smaller or larger, and in whose reference frame?"

'Anti-beings' living in this anti-universe would (just like us) experience time in he direction of entropy increase and therefore experience an expansion (growth with time). They might come to the conclusion that there is a cosmic bounce, and that an anti-universe to theirs might exist. That 'anti-universe' is ours. Now, as I suggested in the article, it might still be that there is only one universe: if the bounce acts as a 'PCT-mirror', both universes are each others mirror images, and therefore there is only one independent universe.

I think there may be different trains of thought at work here, I am not sure which is being discussed, and I apologize for not being more clear -- but I'm not sure the concept of "time" is generally clear.

One train of thought would be that the "forward" motion, in which a moving billiard ball A strikes a stationary billiard ball B, can be "reversed" in the relativistic sense so that ball B is moving, and strikes stationary ball A. Then "forward" and "reversed" depend only on the frame of reference. It seems this kind of symmetric reversibility does not require any arrow of time.

A second train of thought would be that an idealized, special case micro-scale perfectly elastic collision can be reversed in an experiment in the physically real universe. But I am not sure that experiments do show this, since (if I am not mistaken) CP symmetry is violated. Then "time" symmetry must be invoked to preserve some overall universal CPT symmetry, which requires reversing the entire universe, leading to the train of thought I perceive in your analogy.

The third train of thought is that the physically real universe evolves in only one "time" direction -- giving rise to the concept of an "arrow of time". Applying "universally applicable" relativistic reversibility has no effect on the irreversible "time arrow" evolution of systems in the physically real universe. And in the physically real universe, the apparent impossibility of undoing what is already done gives rise to the idea of order leading to disorder (which is not quite as apparent as the impossibility of undoing what is already done, since gravity and electromagnetism appear to create order out of disorder).

A fourth train of thought is that the constant-energy universe evolves from a "big bang" or "bounce" moment of extreme heat and energy density which is treated as being very orderly (although that "order" seems to be a debatable concept even among physicists), and proceeds toward maximum disorder (with the caveat of non-understood dark energy which does not appear to be a constant quantity if more space is always created). Included in this train is the assumption that all the constant energy of the universe was "released", or "appeared" (or "bounced") at this moment, and the "arrow of time" traces the inexorable dilution (or black hole processing) of the constant energy of the creation moment.

I would add a fifth train of thought, which is that the creation or bounce moment (found by running the film backward in time) is an *assumption* which is a necessary foundation to produce an "arrow of time" running with increasing disorder from past to future.

However if the energy of the universe is not a constant quantity but a "constant time" quantity (that is, constantly received from the future in the same way that a signal, e.g. a ray of light, can be treated as being received from the future) then the "arrow of time" can be said to run from future to present (and there is no physically real past). With that in mind, I asked whether the "arrow of time" is understood conventionally as a physically real phenomenon or as a subjective concept.

So the "big bang" or "bounce" idea treats the universe as an isolated system, subject to the second law, moving irreversibly "forward" in a time arrow from past order to future disorder; but there is the alternative that the universe is open, and receives energy from the future -- which however cannot be located in physical reality, but which powers it! Of course this requires a wholesale revision of some "fundamental" concepts.

Nice analogy. So, as I understand it, the arrow of time is based on teh fact that states labelled as earlier cannot contain information on times labelled as later because they do not have enough room to store that information (less digits).

Time then becomes and illusion. All of time already exists, but at every point, there is only a pointer "backwards" to "smaller" times. Larger times are invisible. At the bounce, there are no pointers back, so there is no time before the bounce. There is only a time "after" the bounce, in two directions.

Such a model would mean that there are an infinitesimal number of "me's" that each remember earlier ones (smaller time) but not later ones (bigger time). But there would not an actual evolution or running of time.
(note that this looks a lot like the treatment of time by Bertrand Russell in "Mysticism and Logic and Other Essays", p42 Time)

Don't know this book from Bertrand Russell, so can't comment on the similarities between the above blogpost and his work.

I deliberately didn't direct the discussion into the question "what is time?" The article was already vague enough! I focussed the article on what I think is the key take-away point: the observed link between unbounded reversible dynamics on the one hand, and entropy bounces on the other hand, is not restricted to the class of simple models mentioned in the above. Key claim here is that any unbounded reversible dynamics will have a low entropy bounce.

I need to think more about the one-way memory effects, and might come back in the future to the points you mentioned.

I put your model universe rules into a spreadsheet to see for myself that there is a Big Bounce. Then, as the "bounced" universe's phase space numbers got bigger and bigger, I started to wonder, what happens in that direction? Do we end up with some sort of Big Rebound, once the numbers are large enough? By "large enough", I think I mean, when the numbers describe the maximum possible states.

The growing of the numbers will continue until... your spreadsheet hits upon an integer overflow.
Assuming the 'cosmic spreadsheet' that controls our universe knows of no size limitations, the growth will continue indefinitely.

And .. would that possibly seem to be another kind of zeroing out? The first kind of zeroing out is when we track the numbers in your model universe backwards in time, from which we get a Bounce. This second kind is when the register overflows and all the bits are reset to zero.

This makes me wonder whether or not a bit requires an it in order to manifest. In other words, can local growth (of bits) continue indefinitely if there is not an endless local supply (of its)?

Johannes Koelman wrote: "It was Rudolf Clausius, the inventor of the term 'entropy', who already told us that the entropy of the universe* can go only one direction: it can only increase."

He did not "tell us" that; rather, he DEDUCED it from two FALSE premises. If I am right (that is, if the premises are really false), the law of entropy increase (so disgustingly worshipped by Eddington) is just as meaningful as the statement "The greenness of the crocodile exceeds its length".

Pentcho -- it was Boltzmann, not Clausius, who deduced the second law. And yes, he initially didn't fully appreciate the fundamental assumption behind his Stosszahl ansatz (also referred to s molecular chaos assumption), but this got cleared up later.
I don't think you are doing justice to Eddington's profound insights and neither to the meaningfulness of the 2nd law...!

Johannes Koelman wrote: "Pentcho -- it was Boltzmann, not Clausius, who deduced the second law."

First Carnot deduced the second law in 1824 but used a false premise, then in 1850 Clausius rededuced it from a true premise (heat never flows spontaneously from cold to hot) but the deduction was INVALID:

http://www.mdpi.org/lin/clausius/clausius.htm
"Ueber die bewegende Kraft der Wärme", 1850, Rudolf Clausius: "Carnot assumed, as has already been mentioned, that the equivalent of the work done by heat is found in the mere transfer of heat from a hotter to a colder body, while the quantity of heat remains undiminished. The latter part of this assumption--namely, that the quantity of heat remains undiminished--contradicts our former principle, and must therefore be rejected... (...) It is this maximum of work which must be compared with the heat transferred. When this is done it appears that there is in fact ground for asserting, with Carnot, that it depends only on the quantity of the heat transferred and on the temperatures t and tau of the two bodies A and B, but not on the nature of the substance by means of which the work is done. (...) If we now suppose that there are two substances of which the one can produce more work than the other by the transfer of a given amount of heat, or, what comes to the same thing, needs to transfer less heat from A to B to produce a given quantity of work, we may use these two substances alternately by producing work with one of them in the above process. At the end of the operations both bodies are in their original condition; further, the work produced will have exactly counterbalanced the work done, and therefore, by our former principle, the quantity of heat can have neither increased nor diminished. The only change will occur in the distribution of the heat, since more heat will be transferred from B to A than from A to B, and so on the whole heat will be transferred from B to A. By repeating these two processes alternately it would be possible, without any expenditure of force or any other change, to transfer as much heat as we please from a cold to a hot body, and this is not in accord with the other relations of heat, since it always shows a tendency to equalize temperature differences and therefore to pass from hotter to colder bodies."

The version "entropy always increases" was deduced later (by Clausius again) - see a nice analysis in Jos Uffink's paper:

http://philsci-archive.pitt.edu/archive/00000313/
p.39: "A more important objection, it seems to me, is that Clausius bases his conclusion that the entropy increases in a nicht umkehrbar [irreversible] process on the assumption that such a process can be closed by an umkehrbar [reversible] process to become a cycle. This is essential for the definition of the entropy difference between the initial and final states. But the assumption is far from obvious for a system more complex than an ideal gas, or for states far from equilibrium, or for processes other than the simple exchange of heat and work. Thus, the generalisation to all transformations occurring in Nature is somewhat rash."

"But the assumption is far from obvious for a system more complex than an ideal gas, or for states far from equilibrium, or for processes other than the simple exchange of heat and work. Thus, the generalisation to all transformations occurring in Nature is somewhat rash.""

This conclusion can be attached to each and any major scientific breakthrough in history, from Galileo and Newton, to Darwin and Maxwell, to Einstein, and Schrödinger.

A great scientist is able to generalize the right concepts over incomplete data. As there has never, ever been found a process that violates the second law, and each and every sound theory of thermodynamics honors the second law, this generalization was justified.

In science, it is completely irrelevant whether or not the "original discoverer" made grave errors. Genetics does not have to be scrapped because Mendel's assistants cheated with counting. The only thing that counts is that the discovery holds up under scrutiny.

http://philsci-archive.pitt.edu/archive/00000313/
p.39: "A more important objection, it seems to me, is that Clausius bases his conclusion that the entropy increases in a nicht umkehrbar [irreversible] process on the assumption that such a process can be closed by an umkehrbar [reversible] process to become a cycle."

See no problem? Does this sound like "Clausius bases his conclusion that the entropy increases on the assumption that any irreversible process can be reversed"? Is it difficult to find counterexamples? Irreversible processes that CANNOT be reversed?

"First Carnot deduced the second law in 1824 but used a false premise, then in 1850 Clausius rededuced it from a true premise"

You use 'deduce' in a way that is different from what I am used to. For me, Boltzmann deduced the second law by combining Newton's dynamics laws with statistical reasoning. Clausius inferred from observations that there is a quantity (for which he coined the term 'entropy') that never decreases. Clausius had no clue about the true meaning of entropy at microscales.

Too complicated Dr. Watson! Simple Watson, simple!
In his New Paradigm, Autodynamics, by Dr. R. L. Carezani, he show, without any doubt , that Entropy is CONSTANT and founded in MASS Decay-Energy Absorption that he elevated to a Universal Law, which maintain the Universe in Perpeptual Evolution.
See the whole Theory in the Blog: http://autodynamicslborg.blogspot.com/

Entropy simply is the information content of a physical system, the number of digits needed to fully describe the system down to its smallest details. So the second law tells us that the number of digits needed to describe the universe keeps increasing.

I loved your Fibonacci universe, Johannes, much food for thought.

However, I believe you got the above part backward. Shannon defined information content as the negative of entropy. So, higher-entropy configurations require fewer digits for a full description.

This makes sense not only in statistics but in cosmology as well. Today's universe, full of diverse structures, requires a lot of time and digits to describe. As friction, erosion, and general heat death take their toll on these structures, the universe will become more homogeneous, more like the 2nd sequence above, and will be describable by fewer digits. In the limit, as the last proton in the universe decays, there will be no way to encode any digits at all.

Ok Fred, let's see if I can convince you that you got it backward... :)

Shannon defined entropy as the average 'surprise value' of a stream of symbols. A symbol Pi that occurs in this stream with probability Pi has 'surprise value' log(1/Pi). Averaged over all symbols, the entropy then is H = Sum(i) Pi log(1/Pi).

Now let's apply this to the two sequences you provided. The first sequence contains five symbols that each occur once, i.e. with probability 0.2 and surprise value log(5). This sequence therefore has entropy H = log(5).
Now the second sequence. This has one symbol that occurs with probability 1, and surprise value log(1) = 0. hence this sequence has entropy H = 0, which is the minimum entropy possible. Sequences like (2, 2, 2, 2, 2) that can be described with few symbols "all 2's" are low entropy sequences.

On the fate of our universe: our current universe indeed contains a lot of structure and requires a lot of digits to be described completely. The ultimate fate of the universe is lots and lots of photons spit out by supermassive black holes in the form of Hawking radiation. This universe might seem homogeneous, but contains a lot of structure at fine scales. You need an overwhelming number of digits to describe this universe! ("one photon with wavelength x at spacetime position y, another photon with wavelength z at ...").

I'd never argue cosmology with you, as I know nothing about it. However, there is something wrong with the entropy calculation. I think we have different notions of the state space. I'll think about it and reply again when I get back home to San Diego in a few days.

Glad you put 'surprise' in quotation marks. Others have attempted to use entropy as a measure of our 'ignorance' of the state of a system. Entropy cannot measure psychological states!

"The whole universe expands, so probably its information content increases as well."

I totally agree with that Johannes Koelman. But you believe also in an early universe ("Big Bang theory") which started small and apparently would have a "LOW" entropy.

A small (starting) universe in this view would have to start with a lower amount of information regarding to a later time period. Yet it is incredibly hot and disorderly. Showing all the characteristics of what we call disorder.

One can contract a gas and one can expand it. At what point there is order/disorder and low/high entropy in your opinion?

The Big Bang theory predicts that the 'orderly' state is at a point of highest contraction/smallest volume.
The problem is not the Second Law in my opinion.

You will surely prove me wrong - but I always thought the concept of entropy comes from the times when people tried to get the maximum work out of caloric machines (not that that has changed) and found out its not easy and the term is therefore a purely subjective human concept -ie: if you want movement, heat is a wase - if you want heat, movement is a wase; if you like 2s, 2222 is low - if you like 1s and 3s, 2222 is high; 1415926 may look random to some - others will think its the decimals of pi - all states are equal and it only depends on human communication skills on how to describe each individual one of them - the states change from one to the other but none can be lost - the first "real" law automatically proves the second wrong

I'm not I understand how this links to the Mikado universe idea. If matter attracts to each other to maximize entropy, wouldn't the starting point of the big bang/bounce be the maximum entropy, it would have the highest density of matter? Or would it be the lowest entropy because there are no degrees of freedom?

I really like both of these concepts (big bounce and entropic gravity), I just need help to link them together. In my mind I'm picturing a large sphere (holographic screen) containing the information of the universe and projecting it in to the center. At the stage before the bounce the screen would be full of information (max entropy), and when it's hits the "full" stage it starts to deconstruct everything and free up the screen space until it gets back to the anti bounce point (an extremely dense nugget of hydrogen or whatever).

And gravity would basically be the screen clumping information together the same way you defrag a hard drive? Or the way you have to lay blocks in Testris so as to leave no empty spaces.

I think you should take into consideration the chemistry analogy. Homogeneous means disordered, for a solution: in fact, you can take the system and rearrange it in many ways, without producing observable differences: for example, a water/sugar mixture will be the same after you shake it one, two, or ten times. A living cell or a human body are very low entropy systems, and you can't rearrange them many ways around, without producing a consistently different system as a result - a non-living cell, a man with his stomach in place of his brain, etc.

Entropy is directly proportional to information and, as such, a system wich is capable of delivering great information, such as (1, 8, 4, 3, 6), has higher entropy than (2, 2, 2, 2, 2), delivering no information whatsoever, thus resulting harder to encode, too - you need every single one of the five digits to transfer complete information.

Stephen Wolfram disputes the second law of thermodynamics quite compellingly in "A New Kind of Science," his impressive tome on Complexity Theory. He says it's an artifact of experimenters' bias in that investigators always begin observation at a point of minimum entropy; so of course it seems always to increase.

I think some of the confusion people feel is that the expanding universe just creates a bigger state space to play in: the universe isn't old enough to be in thermal equilibrium yet so, although its entropy is higher than it was yesterday, it is significantly less than it could be. This is the terrible mistake that Hawking apparently (can't remember where I saw it) made at one point - which even I could see was wrong- he apparently thought that if the universe starts contracting again, the thermodynamic arrow will go into reverse immediately. Well there might be some deep link he forgot to mention but in general that's not the way it works: time- symmetry is not restored by simple bounce models. You must consider the eqilibrium ensemble and calculate the rate at which heat-dead, "scrap heap" universes fluctuate back towards the BB - or the Big Suck if you prefer. Pretty small you might think. Also raises the interesting possibility that "we" are not heading towards heat death at all but heading away from it... and very, very likely to turn round any moment. That kind of makes the BB very improbable to be in our past at all. It's much more likely we emerged out of heat-death chaos!

To complicated, totally irrelevant and FLOWED because the Big Bang is a TALE for children together with “”Universes”” and many other fantastic speculations without any scientific basis.
The New Paradigm in Physics-Cosmology, Autodynamics by short, elevated the Mass Decay-Energy Absorption to a fundamental NATURAL LAW that prove that the “Second Law” (of Thermodynamics) IS NOT EXISTENT
AUTODYNAMICS ENTROPY for Laymanhttp://autodynamicslborg.blogspot.com/2011/05/blog-post_30.html

I should add that since the arrow of time only exists for non-equilibrium systems, it is rather academic whether all the universes hanging off the BB are time-reversed with respect to us. I mean take two other universes - each one glares at the other and says "you're time-reversed". We look at them both and feel very smug because we know they are both time-reversed... until it dawns on us that they are also time-reversed with respect to each other. It's all nonsense. Time points away from the low entropy state for everyone.

Mind you, I seriously wonder whether the BB is low entropy at all. After all, the generation of all that extra phase space to play around in came at the cost of an interaction with an inflation field - a plain Boltzmann (?) fluctuation just produces ephemeral bubbles that instantly recollapse as far as I know, don't they? But to make our expanding universe, low entropy gets imposed on a vast amount of matter and energy. That ain't gonna happen if the matter is created in a small space as it's all going to disappear into a black hole PDQ. Hence the magic reversal of gravity has a second role- get those particles away from each other which puts the mass-gravity configuration into a low entropy state once gravity is restored to attraction.

Now the point of this is that the physics is more-or-less comprehensible at this point and the statistical basis for entropy demands that even if there's a magic gravity reversal that creates low entropy, it still needs to be paid for - my guess is the inflation field foots the bill and gets degraded.

Oh I can hear the cries of "crackpot" already! No I don't profess to have a theory, but if you're going to talk about entropy you had better do it all the way through. Or at least for as long as you reckon the laws of physics will be time-symmetric. Once you ditch that, anything goes.